Dimensionally regulated one-loop box scalar integrals with massless internal lines

Using the Feynman parameter method, we have calculated in an elegant manner a set of one-loop box scalar integrals with massless internal lines, but containing 0, 1, 2, or 3 external massive lines. To treat IR divergences (both soft and collinear), the dimensional regularization method has been employed. The results for these integrals, which appear in the process of evaluating one-loop -point integrals and in subdiagrams in QCD loop calculations, have been obtained for arbitrary values of the relevant kinematic variables and are presented in a simple and compact form. Received: 8 March 2001 / Published online: 18 May 2001     

Dimensionally regularized one-loop tensor integralswith massless internal particles

A set of one-loop vertex and box tensor integrals with massless internal particles has been obtained directly without any reduction method to scalar integrals. The results with one or two massive external lines for the vertex integral and zero or one massive external lines for the box integral are shown in this report. Dimensional regularization is employed to treat any soft and collinear (IR) divergence. A series expansion of tensor integrals with respect to an extra space-time dimension for the dimensional regularization is also given. The results are expressed by very short formulas in a manner suitable for a numerical calculation. Arrival of the final proofs: 25 November 2005     

The Massless Higher-Loop Two-Point Function

We introduce a new method for computing massless Feynman integrals analytically in parametric form. An analysis of the method yields a criterion for a primitive Feynman graph G to evaluate to multiple zeta values. The criterion depends only on the topology of G, and can be checked algorithmically. As a corollary, we reprove the result, due to Bierenbaum and Weinzierl, that the massless 2-loop 2-point function is expressible in terms of multiple zeta values, and generalize this to the 3, 4, and 5-loop cases. We find that the coefficients in the Taylor expansion of planar graphs in this range evaluate to multiple zeta values, but the non-planar graphs with crossing number 1 may evaluate to multiple sums with 6^th roots of unity. Our method fails for the five loop graphs with crossing number 2 obtained by breaking open the bipartite graph K _3,4 at one edge. CNRS.     

Moments of heavy quark correlators with two masses: Exact mass dependence to three loops

We compute moments of non-diagonal correlators with two massive quarks. Results are obtained where no restriction on the ratio of the masses is assumed. Both analytical and numerical methods are applied in order to evaluate the two-scale master integrals at three loops. We provide explicit results for the latter which are useful for other calculations. As a by-product we obtain results for the electroweak ρ parameter up to three loops which can be applied to a fourth generation of quarks with arbitrary masses.     

Double-real corrections at $${{\mathcal {O}}(\alpha \alpha _s)}\,$$to single gauge boson production

We consider the \({{\mathcal {O}}(\alpha \alpha _s)}\,\)corrections to single on-shell gauge boson production at hadron colliders. We concentrate on the contribution of all the subprocesses where the gauge boson is accompanied by the emission of two additional real partons and we evaluate the corresponding total cross sections. The latter are divergent quantities, because of soft and collinear emissions, and are expressed as Laurent series in the dimensional regularization parameter. The total cross sections are evaluated by means of reverse unitarity, i.e. expressing the phase-space integrals in terms of two-loop forward box integrals with cuts on the final-state particles. The results are reduced to a combination of master integrals, which eventually are evaluated in terms of generalized polylogarithms. The presence of internal massive lines in the Feynman diagrams, due to the exchange of electroweak gauge bosons, causes the appearance of 14 master integrals which were not previously known in the literature and have been evaluated via differential equations.     

Evaluating massive planar two-loop tensor vertex integrals

Using the parallel/orthogonal space method, we calculate the planar two-loop three-point diagram and two rotated reduced planar two-loop three-point diagrams. Together with the crossed topology, these diagrams are the most complicated ones in the two-loop corrections necessary, for instance, for the decay of the Z⁰ boson. Instead of calculating particular decay processes, we present a new algorithm which allows us to perform arbitrary next-to-next-to-leading order (NNLO) calculations for massive planar two-loop vertex functions in the general mass case. All integration steps up to the last two are performed analytically and will be implemented under xloops as part of the Mainz xloops-GiNaC project. The last two integrations are done numerically using methods like VEGAS and Divonne. Thresholds originating from Landau singularities are found and discussed in detail. In order to demonstrate the numerical stability of our methods we consider particular Feynman integrals which contribute to different physical processes. Our results can be generalized to the case of the crossed topology.     

Configuration space based recurrence relations for sunset-type diagrams

We derive recurrence relations for the calculation of multiloop sunset-type diagrams with large powers of massive propagators. The technique is formulated in configuration space and exploits the explicit form of the massive propagator raised to a given power. We write down and evaluate a convenient set of basis integrals. The method is well suited for a numerical evaluation of this class of diagrams. We give explicit analytical formulae for the basis integrals in the asymptotic regime. Received: 30 April 1999 / Revised version: 2 July 1999 / Published online: 14 October 1999     

Gauge boson exchange in AdSd+1

We study the amplitude for exchange of massless gauge bosons between pairs of massive scalar fields in anti-de Sitter space. In the AdS/CFT correspondence this amplitude describes the contribution of conserved flavor symmetry currents to 4-point functions of scalar operators in the boundary conformal theory. A concise, covariant, Y2K compatible derivation of the gauge boson propagator in AdS_{d + 1} is given. Techniques are developed to calculate the two bulk integrals over AdS space leading to explicit expressions or convenient, simple integral representations for the amplitude. The amplitude contains leading power and sub-leading logarithmic singularities in the gauge boson channel and leading logarithms in the crossed channel. The new methods of this paper are expected to have other applications in the study of the Maldacena conjecture.     

Calculating contracted tensor Feynman integrals

A recently derived approach to the tensor reduction of 5-point one-loop Feynman integrals expresses the tensor coefficients by scalar 1-point to 4-point Feynman integrals completely algebraically. In this Letter we derive extremely compact algebraic expressions for the contractions of the tensor integrals with external momenta. This is based on sums over signed minors weighted with scalar products of the external momenta. With these contractions one can construct the invariant amplitudes of the matrix elements under consideration, and the evaluation of one-loop contributions to massless and massive multi-particle production at high energy colliders like LHC and ILC is expected to be performed very efficiently.     

Master integrals for four-loop massless propagators up to weight twelve

We evaluate a Laurent expansion in dimensional regularization parameter ?=(4−d)/2 of all the master integrals for four-loop massless propagators up to weight twelve, using a recently developed method of one of the present coauthors (R.L.) and extending thereby results by Baikov and Chetyrkin obtained at weight seven. We observe only multiple zeta values in our results. Therefore, we conclude that all the four-loop massless propagator integrals, with any integer powers of numerators and propagators, have only multiple zeta values in their epsilon expansions up to weight twelve.     

Four loop massless propagators: An algebraic evaluation of all master integrals

The old “glue-and-cut” symmetry of massless propagators, first established in Ref. [1] (Chetyrkin and Tkachov, 1981), leads — after reduction to master integrals is performed   — to a host of non-trivial relations between the latter. The relations constrain the master integrals so tightly that they all can be analytically expressed in terms of only few, essentially trivial, watermelon-like integrals. As a consequence we arrive at explicit analytical results for all master integrals appearing in the process of reduction of massless propagators at three and four loops. The transcendental structure of the results suggests a clean explanation of the well-known mystery of the absence of even zetas (ζ_2n${\zeta }_{2n}$) in the Adler function and other similar functions essentially reducible to massless propagators. Once a reduction of massless propagators at five loops is available, our approach should be also applicable for explicitly performing the corresponding five-loop master integrals.     

Dispersive Calculation of the Massless Multi-loop Sunrise Diagram

The massless sunrise diagram with an arbitrary number of loops is calculated in a simple but formal manner. The result is then verified by rigorous mathematical treatment. Pitfalls in the calculation with distributions are highlighted and explained. The result displays the high energy behaviour of the massive sunrise diagrams, whose calculation is involved already for the two-loop case.     

Massless and massive one-loop three-point functions in negative dimensional approach

In this article we present the complete massless and massive one-loop triangle diagram results using the negative dimensional integration method (NDIM). We consider the following cases: massless internal fields; one massive, two massive with the same mass m and three equal masses for the virtual particles. Our results are given in terms of hypergeometric and hypergeometric-type functions of the external momenta (and masses for the massive cases) where the propagators in the Feynman integrals are raised to arbitrary exponents and the dimension of the space-time is D. Our approach reproduces the known results; it produces other solutions as yet unknown in the literature as well. These new solutions occur naturally in the context of NDIM revealing a promising technique to solve Feynman integrals in quantum field theories. Received: 14 April 2002 / Revised version: 18 July 2002 / Published online: 7 October 2002 RID="a" ID="a" e-mail: suzuki@ift.unesp.br RID="b" ID="b" e-mail: esdras@ift.unesp.br RID="c" ID="c" e-mail: schmidt@fisica.ufpr.br     

Nine-propagator master integrals for massless three-loop form factors

We complete the calculation of master integrals for massless three-loop form factors by computing the previously-unknown three diagrams with nine propagators in dimensional regularisation. Each of the integrals yields a six-fold Mellin–Barnes representation which we use to compute the coefficients of the Laurent expansion in ?. Using Riemann ζ functions of up to weight six, we give fully analytic results for one integral; for a second, analytic results for all but the finite term; for the third, analytic results for all but the last two coefficients in the Laurent expansion. The remaining coefficients are given numerically to sufficiently high accuracy for phenomenological applications.     

Analytic two-loop master integrals for tW production at hadron colliders: I

We present the analytic calculation of two-loop master integrals that are relevant for tW production at hadron colliders. We focus on the integral families with only one massive propagator. After selecting a canonical basis, the differential equations for the master integrals can be transformed into the d ln form. The boundaries are determined by simple direct integrations or regularity conditions at kinematic points without physical singularities. The analytical results in this work are expressed in terms of multiple polylogarithms, and have been checked via numerical computations.     

Reduction method for dimensionally regulatedone-loop <Emphasis Type="Italic">N</Emphasis>-point Feynman integrals

We present a systematic method for reducing an arbitrary one-loop N-point massless Feynman integral with generic 4-dimensional momenta to a set comprised of eight fundamental scalar integrals: six box integrals in D = 6, a triangle integral in D = 4, and a general two-point integral in D space-time dimensions. All the divergences present in the original integral are contained in the general two-point integral and associated coefficients. The problem of vanishing of the kinematic determinants has been solved in an elegant and transparent manner. Being derived with no restrictions regarding the external momenta, the method is completely general and applicable for arbitrary kinematics. In particular, it applies to the integrals in which the set of external momenta contains subsets comprised of two or more collinear momenta, which are unavoidable when calculating one-loop contributions to the hard-scattering amplitude for exclusive hadronic processes at large-momentum transfer in PQCD. The iterative structure makes it easy to implement the formalism in an algebraic computer program.Received: 18 August 2003, Revised: 6 February 2004, Published online: 23 April 2004     

Multi-loop Feynman integrals and conformal quantum mechanics

New algebraic approach to analytical calculations of D-dimensional integrals for multi-loop Feynman diagrams is proposed. We show that the known analytical methods of evaluation of multi-loop Feynman integrals, such as integration by parts and star-triangle relation methods, can be drastically simplified by using this algebraic approach. To demonstrate the advantages of the algebraic method of analytical evaluation of multi-loop Feynman diagrams, we calculate ladder diagrams for the massless φ³ theory. Using our algebraic approach we show that the problem of evaluation of special classes of Feynman diagrams reduces to the calculation of the Green functions for specific quantum mechanical problems. In particular, the integrals for ladder massless diagrams in the φ³ scalar field theory are given by the Green function for the conformal quantum mechanics.     

Differential reduction of generalized hypergeometric functions from Feynman diagrams: One-variable case

The differential-reduction algorithm, which allows one to express generalized hypergeometric functions with parameters of arbitrary values in terms of the same functions with parameters whose values differ from the original ones by integers, is discussed in the context of evaluating Feynman diagrams. Where this is possible, we compare our results with those obtained using standard techniques. It is shown that the criterion of reducibility of multiloop Feynman integrals can be reformulated in terms of the criterion of reducibility of hypergeometric functions. The relation between the numbers of master integrals obtained by differential reduction and integration by parts is discussed.     

New relationships between Feynman integrals

New types of relationships between Feynman integrals are presented. It is shown that Feynman integrals satisfy functional equations connecting integrals with different values of scalar invariants and masses. A method is proposed for obtaining such relations. The derivation of functional equations for one-loop propagator- and vertex-type integrals is given. It is shown that a propagator-type integral can be written as a sum of two integrals with modified scalar invariants and one propagator massless. The vertex-type integral can be written as a sum over vertex integrals with all but one propagator massless and one external momenta squared equal to zero. It is demonstrated that the functional equations can be used for the analytic continuation of Feynman integrals to different kinematic domains.     

The massless supersymmetric ladder with L rungs

We show that in the massless N=1$N=1$ supersymmetric Wess–Zumino theory it is possible to devise a computational strategy by which the x-space calculation of the ladder 4-point correlators can be carried out without introducing any regularization. As an application we derive a representation valid at all loop orders in terms of conformal invariant integrals. We obtain an explicit expression of the 3-loop ladder diagram for collinear external points.